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1.2 the Beginnings of String Theory 9 Durham E-Theses Aspects of the gauged, twisted, SL(2|1)/SL(2|1) Wess-Zumino-Novikov-Witten model Koktava, Rachel-Louise Kvertus How to cite: Koktava, Rachel-Louise Kvertus (1996) Aspects of the gauged, twisted, SL(2|1)/SL(2|1) Wess-Zumino-Novikov-Witten model, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/5291/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Aspects of the Gauged, Twisted, SL(2|1)/SL(2|1) Wess-Zumino-Novikov-Witten Model Aspects of the Gauged, Twisted, SL(2|1)/SL(2|1) Wess-Zumino-Novikov-Witten Model Submitted for the degree of Doctor of Philosophy at The Department of Mathematical Sciences The University of Durham by Rachel-Louise Kvertus Koktava September 1996 The copyright of this thesis rests with the author. No quotation from it should be pubUshed without the written consent of the author and information derived from it should be acknowledged. 3 JUL 199? I dedicate this work to my grandfather Josef Kokta (1920-1992) When your spirt left this world The light went out Of many lives. Acknowledgments A great vote of thanks goes to those who were instrumental in the development of this work, and my education. Of these people Dr. Anne Taormina is prominent. Thanks are owed to her for the willingness to take on a student and together begin the study of an unfamiliar and difficult field. For her patience and guidance, and willingness to help, I say thank you. Of equal standing is Dr. Peter Bowcock, whose open door and broad knowledge were invaluable in allowing me to understand a complex field of study. Professor Ed Corrigan should not be forgotten. Although not taking an active part in the research, he has been willing to read and comment upon drafts of both published material and this thesis. Three more acknowledgments should be made. Firstly to the staif of the De• partment of Mathematical Sciences at Durham University for their pleasant, sup• portive and continuous help. Secondly to Ruth Fraser for support of another kind without which I would have been unable to work, and finally to my funding body, the Particle Physics and Astronomy Research Council — formerly known as the Science and Engineering Research Council. Table of Contents Table of Contents Preface 3 Abstract 4 Introduction 5 Chapter One : Ancestry of the G/G Models 8 1.1 Introduction 8 1.2 The Beginnings of String Theory 9 1.3 Polyakov's Path Integral Formalism 13 1.4 Development of the G/G Models 18 Chapter Two : The SL(2|1) WZNW Model 24 2.1 Introduction 24 2.2 Introducing the Wess-Zumino-Novikov-Witten Model 25 2.3 The Lie Super-Algebra 5L(2|1) 28 2.4 The Wakimoto Construction 33 2.5 The Non-hnear Transformation between Currents 40 2.6 The Quantum Currents 44 2.7 The Quantum Field Transformation 47 Chapter Three : The Gauged, Twisted SL (211)/SL (211) WZNW Model 50 3.1 Introduction 50 3.2 The Gauged WZNW Model 52 3.3 The Twisted G/G Model 58 3.4 The Non-Critical N=2 String 60 3.5 Hamiltonian Reduction 68 Table of Contents 3.6 The Equivalence between the Gauged, Twisted WZNW Model and the Fermionic String 78 3.7 Topological Conformal Field Theories 80 3.8 BRST Cohomology of the Non-Critical String 82 Chapter 4 : The Space of Physical States 87 4.1 Introduction 87 4.2 The Isomorphism Between the Ramond and Neveu-Schwarz Sectors 88 4.3 The BRST Cohomology 94 Conclusions and Speculations 102 Appendix A : Details of the Algebra sl(2|l) 107 Appendix B : Proof of the Field Transformation 110 Appendix C : Modal Decomposition of Currents 115 Appendix D : REDUCE Program used for Calculations 119 References 126 Preface The following work is presented by myself for the degree of Doctor of Philos• ophy at the University of Durham. The content of this thesis is a blend of known work, research undertaken with my supervisor, and research of my own. Only chap• ter one contains no original work. Chapter two is an even split between work of my own and work jointly undertaken with Dr. Anne Taormina and Dr. Peter Bowcock, except where references indicate otherwise. Chapter three is composed of work undertaken jointly with Dr. Taormina and Dr. Bowcock. Chapter four is largely my own work, with some assistance in the calculations from Dr. Taormina. The work in this dissertation does not follow the full rigour that the reader may have anticipated. I make no apologies for this. As someone whose intuition and background lies in the realm of physics, this thesis has been a great edification in study. In attempting to understand the work of mathematicians I have repeat• edly been struck by the difficulty I, as a physicist, have encountered in trying to comprehend what should, and could, be simple. I have therefore specifically set out to keep the nature of this work readily accessible to physicists, postgraduate stu• dents, and the knowledgeable alike. I assume that if the reader is qualified enough to see the failings in the lack of full rigour, then the reader is qualified enough to supply those deficits for themselves. The copyright of this thesis rests with the author. No quotation from it should be published without her prior written consent and information derived from it should be acknowledged. Abstract In this thesis we examine some of the interesting aspects of the Wess-Zumino- Novikov-Witten model when this model has been gauged and its energy tensor twisted by the addition of the derivative of one of its Cartan subalgebra valued currents. Specifically we consider the group valued model with the group taken as SL(2\1) which is the Lie super group used to describe N = 2 supersymmetry. This model is advocated as being a good and natural description of the N =^ 2 superstring (also known as the charged spinning string, or A'^ = 2 fermionic string) when it tensors an additional topological system of ghosts. The evidence for this assertion is presented by gauging and twisting the model and then extracting the N = 2 super Liouville action by the method of Hamiltonian reduction. The connection between the 5L (2|1)/5L (2|1) Wess-Zumino-Novikov-Witten model and field theory is made through its current algebra. As is true of many super groups there exists more than one interpretation of the Dynkin diagram for the algebra of 5L(2|1) and this results in more that one set of currents for this model. The classical and quantum currents in free field form are found in both cases, as is the highly non-linear transformation by which the two sets of currents are related. An analysis of a section of the cohomology of physical states of the model is undertaken. It is shown that the additional topological ghost system that tensors the gauged, twisted SL(2\1) model when it describes the N = 2 string only contributes a vacuum state to the overall cohomology, so reducing the analysis. As the 5L (2|1)/5L (2|1) Wess-Zumino-Novikov-Witten model is a topological field theory its spectrum of physical states he in the cohomology class defined with respect to the BRST charge. The spectrum formed from the free field currents composes the so called Wakimoto module and this is calculated via the BRST formahsm. Introduction Introduction Mathematics is where the answer is right and every• thing is nice and you can look out of the window and see the blue sky — or the answer is wrong and you have to start all over and try again and see how it comes out this time. From Complete Poems by Car] Sandburg It is one of those curious patterns of history which tells us to expect a revolu• tion in physics, on average, every quarter of a century. It is perhaps this expectation that caused many individuals to see the coming of string theory as the overdue fulfilment of this cycle. Whether the theory of strings is the long awaited golden goose of modern the• oretical physics — or the golden duck — is as unknown to us as the future. Yet one Introduction 6 thing is clear. During its development it has shown itself to be vast undiscovered country for both the mathematician and physicist alike as they have struggled to uncover the rich mathematical structures, and have been forced to apply them• selves to the solutions of new and stimulating problems. In recent years the flow of ideas has been stemmed by the sheer complexity of the analysis needed. As a consequence the remaining problems in string theory are likely to remain problems for some time to come. This work is a peep at one such problem — the solving, off the critical dimension, of the Liouville sector for the model of the fermionic string, otherwise known as the A'^ = 2 superstring or the charged spinning string.
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